I'm not sure how things would go if a woman arrived in the UK at 32 weeks pregnant, but I imagine it would involve some tests and multiple appointments fairly quickly. Medical records don't cross countries very well.
As for the US, it depends on where and what hospital, it varies a lot. In some areas, many things are medicalized more, birth being one of them, and in others it's not.
Also, gas & air is used in the US - it's just not part of the labour ward and more used in dentistry or for things like wart removal.
And blaming the US's poorer outcomes on solely the women's action is a great way to ignore the flaws in the system and rings a lot like those US politicians who go on about how the US's maternal and birth outcomes would be great if black women were removed from the stats.
I don't understand the backlash against medicalisation...death in childbirth and stillbirths were just par for the course not that many years ago. I'll take the science, thanks.
Medical abuse is a big reason, something pregnant girls and women are known to be at higher risk for and less likely than the already abysmal typical rates of getting it actually dealt with on either side of the Atlantic, even more for those disabled and/or of marginalized ethnicities (even when they go to all their appointments).
It also plays a major role in the anti-vax crowd and the previously mentioned freebirthers. When someone loses trust that HCPs have their best interest at heart and can get away with anything, and has horrific experience of the system failing to protect them at their most vulnerable, they can go really hard against any type for medicalisation and getting back from that has barriers, often compounded by poor science education that many areas of the US are well known for, but also in parts of the UK.